X-Ray crystallography is the only technique capable of elucidating the three-dimensional structures of biological macromolecules. This technique, however, not only requires suitable crystals of the molecule of interest but the chemical preparation and analysis of a series of related crystal structures as well, the so-called isomorphous derivatives. For this reason the structure determination of macromolecules using the method of multiple isomorphous replacement has been an extremely time consuming process. The proposed research would combine the method of multiple isomorphous replacement with the highly developed direct method of phase determination which has been extremely successful in the ab initio determination of crystal structures of small molecules. In this way, by emphasizing mathematical techniques, rather than the time consuming chemical ones needed for the preparation of many isomorphous derivatives, the structure determination process may be automated and the time required to prepare the smaller number of isomorphous derivatives greatly reduced. It is proposed to implement phasing formulas based on a new analysis of the errors inherent in the isomorphous replacement process, to search for better ways to implement anomalous scattering information, to integrate the direct methods technique with that of isomorphous replacement, and to evaluate the effectiveness of the resulting procedures on real macromolecular data sets.